[{"has_accepted_license":"1","department":[{"_id":"JuFi"}],"date_created":"2023-06-11T22:00:40Z","month":"05","date_published":"2023-05-30T00:00:00Z","article_type":"original","publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"publication":"Foundations of Computational Mathematics","day":"30","type":"journal_article","status":"public","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"isi":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s10208-023-09613-y"}],"ddc":["510"],"doi":"10.1007/s10208-023-09613-y","year":"2023","external_id":{"isi":["000999623100001"]},"title":"Bias in the representative volume element method: Periodize the ensemble instead of its realizations","oa":1,"date_updated":"2023-08-02T06:12:39Z","article_processing_charge":"Yes (via OA deal)","_id":"13129","publication_identifier":{"eissn":["1615-3383"],"issn":["1615-3375"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria).","quality_controlled":"1","oa_version":"Published Version","publication_status":"epub_ahead","citation":{"ama":"Clozeau N, Josien M, Otto F, Xu Q. Bias in the representative volume element method: Periodize the ensemble instead of its realizations. <i>Foundations of Computational Mathematics</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s10208-023-09613-y\">10.1007/s10208-023-09613-y</a>","mla":"Clozeau, Nicolas, et al. “Bias in the Representative Volume Element Method: Periodize the Ensemble Instead of Its Realizations.” <i>Foundations of Computational Mathematics</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s10208-023-09613-y\">10.1007/s10208-023-09613-y</a>.","ista":"Clozeau N, Josien M, Otto F, Xu Q. 2023. Bias in the representative volume element method: Periodize the ensemble instead of its realizations. Foundations of Computational Mathematics.","short":"N. Clozeau, M. Josien, F. Otto, Q. Xu, Foundations of Computational Mathematics (2023).","ieee":"N. Clozeau, M. Josien, F. Otto, and Q. Xu, “Bias in the representative volume element method: Periodize the ensemble instead of its realizations,” <i>Foundations of Computational Mathematics</i>. Springer Nature, 2023.","apa":"Clozeau, N., Josien, M., Otto, F., &#38; Xu, Q. (2023). Bias in the representative volume element method: Periodize the ensemble instead of its realizations. <i>Foundations of Computational Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10208-023-09613-y\">https://doi.org/10.1007/s10208-023-09613-y</a>","chicago":"Clozeau, Nicolas, Marc Josien, Felix Otto, and Qiang Xu. “Bias in the Representative Volume Element Method: Periodize the Ensemble Instead of Its Realizations.” <i>Foundations of Computational Mathematics</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s10208-023-09613-y\">https://doi.org/10.1007/s10208-023-09613-y</a>."},"abstract":[{"text":"We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior ahom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨⋅⟩ and the corresponding linear elliptic operator −∇⋅a∇. In line with the theory of homogenization, the method proceeds by computing d=3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨⋅⟩. We make this point by investigating the bias (or systematic error), i.e., the difference between ahom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O(L−1). In case of a suitable periodization of ⟨⋅⟩\r\n, we rigorously show that it is O(L−d). In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨⋅⟩\r\n of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function.","lang":"eng"}],"author":[{"id":"fea1b376-906f-11eb-847d-b2c0cf46455b","last_name":"Clozeau","full_name":"Clozeau, Nicolas","first_name":"Nicolas"},{"first_name":"Marc","last_name":"Josien","full_name":"Josien, Marc"},{"full_name":"Otto, Felix","last_name":"Otto","first_name":"Felix"},{"first_name":"Qiang","full_name":"Xu, Qiang","last_name":"Xu"}]},{"day":"17","type":"journal_article","status":"public","publication":"Foundations of Computational Mathematics","file_date_updated":"2021-12-13T15:47:54Z","month":"08","article_type":"original","date_published":"2021-08-17T00:00:00Z","scopus_import":"1","publisher":"Springer","language":[{"iso":"eng"}],"has_accepted_license":"1","department":[{"_id":"MaMo"}],"date_created":"2021-11-03T10:59:08Z","file":[{"success":1,"relation":"main_file","content_type":"application/pdf","creator":"alisjak","file_id":"10542","file_size":2305731,"file_name":"2021_Springer_Mondelli.pdf","checksum":"9ea12dd8045a0678000a3a59295221cb","date_created":"2021-12-13T15:47:54Z","access_level":"open_access","date_updated":"2021-12-13T15:47:54Z"}],"citation":{"ama":"Mondelli M, Thrampoulidis C, Venkataramanan R. Optimal combination of linear and spectral estimators for generalized linear models. <i>Foundations of Computational Mathematics</i>. 2021. doi:<a href=\"https://doi.org/10.1007/s10208-021-09531-x\">10.1007/s10208-021-09531-x</a>","mla":"Mondelli, Marco, et al. “Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models.” <i>Foundations of Computational Mathematics</i>, Springer, 2021, doi:<a href=\"https://doi.org/10.1007/s10208-021-09531-x\">10.1007/s10208-021-09531-x</a>.","short":"M. Mondelli, C. Thrampoulidis, R. Venkataramanan, Foundations of Computational Mathematics (2021).","ista":"Mondelli M, Thrampoulidis C, Venkataramanan R. 2021. Optimal combination of linear and spectral estimators for generalized linear models. Foundations of Computational Mathematics.","ieee":"M. Mondelli, C. Thrampoulidis, and R. Venkataramanan, “Optimal combination of linear and spectral estimators for generalized linear models,” <i>Foundations of Computational Mathematics</i>. Springer, 2021.","apa":"Mondelli, M., Thrampoulidis, C., &#38; Venkataramanan, R. (2021). Optimal combination of linear and spectral estimators for generalized linear models. <i>Foundations of Computational Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s10208-021-09531-x\">https://doi.org/10.1007/s10208-021-09531-x</a>","chicago":"Mondelli, Marco, Christos Thrampoulidis, and Ramji Venkataramanan. “Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models.” <i>Foundations of Computational Mathematics</i>. Springer, 2021. <a href=\"https://doi.org/10.1007/s10208-021-09531-x\">https://doi.org/10.1007/s10208-021-09531-x</a>."},"publication_status":"published","abstract":[{"lang":"eng","text":"We study the problem of recovering an unknown signal 𝑥𝑥 given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator 𝑥𝑥^L and a spectral estimator 𝑥𝑥^s. The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine 𝑥𝑥^L and 𝑥𝑥^s. At the heart of our analysis is the exact characterization of the empirical joint distribution of (𝑥𝑥,𝑥𝑥^L,𝑥𝑥^s) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of 𝑥𝑥^L and 𝑥𝑥^s, given the limiting distribution of the signal 𝑥𝑥. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form 𝜃𝑥𝑥^L+𝑥𝑥^s and we derive the optimal combination coefficient. In order to establish the limiting distribution of (𝑥𝑥,𝑥𝑥^L,𝑥𝑥^s), we design and analyze an approximate message passing algorithm whose iterates give 𝑥𝑥^L and approach 𝑥𝑥^s. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately."}],"author":[{"id":"27EB676C-8706-11E9-9510-7717E6697425","orcid":"0000-0002-3242-7020","last_name":"Mondelli","full_name":"Mondelli, Marco","first_name":"Marco"},{"full_name":"Thrampoulidis, Christos","last_name":"Thrampoulidis","first_name":"Christos"},{"full_name":"Venkataramanan, Ramji","last_name":"Venkataramanan","first_name":"Ramji"}],"keyword":["Applied Mathematics","Computational Theory and Mathematics","Computational Mathematics","Analysis"],"arxiv":1,"article_processing_charge":"Yes (via OA deal)","date_updated":"2023-09-05T14:13:57Z","oa":1,"publication_identifier":{"issn":["1615-3375"],"eissn":["1615-3383"]},"_id":"10211","quality_controlled":"1","project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"oa_version":"Published Version","acknowledgement":"M. Mondelli would like to thank Andrea Montanari for helpful discussions. All the authors would like to thank the anonymous reviewers for their helpful comments.","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","year":"2021","doi":"10.1007/s10208-021-09531-x","external_id":{"arxiv":["2008.03326"],"isi":["000685721000001"]},"title":"Optimal combination of linear and spectral estimators for generalized linear models","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"isi":1,"ddc":["510"]}]